After reviewing the general properties of zero-energy quantum states, we give the explicit solutions of the \seq with $E=0$ for the class of potentials $C=|\gamme|/r^{\nu}$, where $-\infty2$, these solutions are normalizable and correspond to bound staes, if the angular momentum quantum number $|>0$. [These states are normalizable, even for $1=0$, if we increase the space dimension, $D$, beyond 4; i.e. for $D>4$.] For $\nu